Classical solutions for the Carroll-Field-Jackiw-Proca electrodynamics

In the present work, we investigate classical solutions of the Maxwell-Carroll-Field-Jackiw-Proca (MCFJP) electrodynamics for the cases of a purely timelike and spacelike Lorentz-violating (LV) background. Starting from the MCFJP Lagrangian and the associated wave equations written for the potential four-vector, the tensor form of the Green function is achieved. In the timelike case, the components of the stationary Green function are explicitly written. The classical solutions for the electric and magnetic field strengths are then evaluated, being observed that the electric sector is not modified by the LV background, keeping the Maxwell-Proca behavior. The magnetic field associated with a charge in uniform motion presents an oscillating behavior that also provides an oscillating MCFJ solution (in the limit of a vanishing Proca mass), but does not recover the Maxwell-Proca solution in the limit of vanishing background. In the spacelike case, the stationary Green function is written and also explicitly carried out in the regime of a small background. The electric and magnetic fields reveal to possess an exponentially decaying behavior, that recover the Maxwell-Proca solutions.

Figure 1

Simultaneous plot for the magnetic field in the radial interval $0.4<r<4$ of the following models: Maxwell-Proca (for $M_A=1$)—circle dotted line; MCFJ—cross dotted line (for $v_0=3$, $M_A=0$); MCFJ-Proca—box dotted line (for $v_0=3$, $M_A=1$), MCFJ-Proca—intermediary continuous line (for $v_0=6$, $M_A=1$), MCFJ-Proca—thicker continuous line (for $v_0=10$, $M_A=1$), and $u=0.5$ in all cases.

Figure 2

Simultaneous plot of the magnetic fields for the MCFJ (dotted curves—$M_A=0$) and MCFJ-Proca (continuous curves) models in the radial interval $0.4<r<3$. The lower two curves—point dotted and continuous superposed line—correspond to ($v_0=2$, $M_A=0$) and ($v_0=2$, $M_A=0.5$). For the two intermediary curves—box dotted and continuous thin line: ($v_0=4$, $M_A=0$), and ($v_0=4$, $M_A=1$). For the two upper curves—cross dotted and thicker continuous line: ($v_0=9$, $M_A=0$) and ($v_0=9$, $M_A=4$). It was used $u=0.5$ in all graphs.